# Distinct types of eigenvector localization in networks

arXiv: https://arxiv.org/abs/1505.06024 · HTML: https://mlsanity.com/paper/1505.06024/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 May 2015.

## Built on by (most cited first, 4 of 4)

- [Grokking modular arithmetic](https://mlsanity.com/paper/2301.02679/) (2023)
- [To grok or not to grok: Disentangling generalization and memorization on corrupted algorithmic datasets](https://mlsanity.com/paper/2310.13061/) (2023)
- [Unveiling Memorization-Generalization Coexistence: A Case Study on Arithmetic Tasks with Label Noise](https://mlsanity.com/paper/2605.18022/) (2026)
- [Stratified Graph Spectra](https://mlsanity.com/paper/2201.03696/) (2022)

## Abstract (arXiv)

The spectral properties of the adjacency matrix provide a trove of information about the structure and function of complex networks. In particular, the largest eigenvalue and its associated principal eigenvector are crucial in the understanding of nodes centrality and the unfolding of dynamical processes. Here we show that two distinct types of localization of the principal eigenvector may occur in heterogeneous networks. For synthetic networks with degree distribution P(q) ~ q-γ, localization occurs on the largest hub if γ>5/2; for γ<5/2 a new type of localization arises on a mesoscopic subgraph associated with the shell with the largest index in the K-core decomposition. Similar evidence for the existence of distinct localization modes is found in the analysis of real-world networks. Our results open a new perspective on dynamical processes on networks and on a recently proposed alternative measure of node centrality based on the non-backtracking matrix.